Lemma 74.42.8. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $\mathcal{G}$ be a coherent $\mathcal{O}_ X$-module, $(\mathcal{F}_ n)$ an object of $\textit{Coh}(X, \mathcal{I})$, and $\alpha : (\mathcal{F}_ n) \to \mathcal{G}^\wedge $ a map whose kernel and cokernel are annihilated by a power of $\mathcal{I}$. Then there exists a unique (up to unique isomorphism) triple $(\mathcal{F}, a, \beta )$ where

$\mathcal{F}$ is a coherent $\mathcal{O}_ X$-module,

$a : \mathcal{F} \to \mathcal{G}$ is an $\mathcal{O}_ X$-module map whose kernel and cokernel are annihilated by a power of $\mathcal{I}$,

$\beta : (\mathcal{F}_ n) \to \mathcal{F}^\wedge $ is an isomorphism, and

$\alpha = a^\wedge \circ \beta $.

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